User:Marsupilamov/Dold-Thom theorem
In topology, a branch of mathematics,
Let (X,e) be a pointed topological space.
Reduced product of a space
[edit]The James reduced product J(X) is the quotient of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xk−1,e,xk+1,...,xn) with (x1,...,xk−1, xk+1,...,xn). In other words, its underlying set is the free monoid generated by X (with unit e). It was introduced by Ioan James (1955).
For a connected CW complex X, the James reduced product J(X) has the same homotopy type as ΩΣX, the loop space of the suspension of X.
Infinite symmetric product
[edit]The commutative analogue of the James reduced product is called the infinite symmetric product.
Definition
[edit]The infinite symmetric product SP(X) is the quotient of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xk−1,e,xk+1,...,xn) with (x1,...,xk−1, xk+1,...,xn) and identifying any point with any other point given by permuting its coordinates. In other words, its underlying set is the free commutative monoid generated by X (with unit e), and is the abelianization of the James reduced product.
The Dold-Thom theorem
[edit]The infinite symmetric product appears in the Dold–Thom theorem, proved by Albrecht Dold and René Thom (1956, 1958). It states that the homotopy group πi(SP(X)) of the infinite symmetric product SP(X) of X is the homology Hi(X,Z) of the singular complex of the suspension of X.
References
[edit]- Dold, Albrecht; Thom, René (1956), "Une généralisation de la notion d'espace fibré. Application aux produits symétriques infinis", Les Comptes rendus de l'Académie des sciences, 242: 1680–1682, MR 0077121
- Dold, Albrecht; Thom, René (1958), "Quasifaserungen und unendliche symmetrische Produkte", Annals of Mathematics. Second Series, 67: 239–281, ISSN 0003-486X, JSTOR 1970005, MR 0097062
- James, I. M. (1955), "Reduced product spaces", Annals of Mathematics. Second Series, 62: 170–197, ISSN 0003-486X, JSTOR 2007107, MR 0073181
- John Baez' This week's finds on the Thom-Dold
theorem